Past

We compute the asymptotics of Arakelov functions if smooth curves degenerate to semistable singular curves. The motivation was to determine whether the delta function defines a metric on the boundary of moduli space. In fact things are slightly more complicated. The main result states that the asymptotics is mostly governed by the graph associated to the degeneration, with some subleties. The topic has been also treated by R. deJong and my student R. Wilms.

A cohomology class on the diffeomorphism group Diff(M) of a manifold M

can be thought of as a characteristic class for smooth M-bundles.
I will survey a technique for producing examples of such classes,
and then explain how the signature (of 4-manifolds) provides an
obstruction to this technique in dimension 3.

I will define Miller-Morita-Mumford classes and explain how we can
think of them as coming from classes on the cobordism category.
Madsen and Weiss showed that for a surface S of genus g all cohomology
classes
of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.
This technique has been successfully ported to higher even dimensions d= 2n,
but it cannot possibly work in odd dimensions:
a theorem of Ebert says that for d=3 all MMM-classes are trivial.
In the second part of my talk I will sketch a new proof of (a part of)
Ebert's theorem.
I first recall the definition of the signature sign(W) of a 4 manifold W,
and some of its properties, such as additivity with respect to gluing.
Using the signature and an idea from the world of 1-2-3-TQFTs,
I then go on to define a 'central extension' of the three dimensional
cobordism category.
This central extension corresponds to a 2-cocycle on the 3d cobordism
category,
and we will see that the construction implies that the associated MMM-class
has to vanish on all 3-dimensional manifold bundles.

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

I will review the equivalence of categories of a Bernstein component of a p-adic classical group with the category of right modules over a certain affine Hecke algebra (with parameters) that I obtained previously. The parameters can be made explicit by the parametrization of supercuspidal representations of classical groups obtained by C. Moeglin, using methods of J. Arthur. Via this equivalence, I can show that the category of smooth complex representations of a quasisplit $p$-adic classical group and its pure inner forms is naturally decomposed into subcategories that are equivalent to the tensor product of categories of unipotent representations of classical groups (in the sense of G. Lusztig). All classical groups (general linear, orthogonal, symplectic and unitary groups) appear in this context.
 

The Landau-DeGennes Q-model of uniaxial nematic liquid crystals seeks a rank-one

traceless tensor Q that minimizes a Frank-type energy plus a double well potential

that confines the eigenvalues of Q to lie between -1/2 and 1. We propose a finite

element method (FEM) which preserves this basic structure and satisfies a discrete

form of the fundamental energy estimates. We prove that the discrete problem Gamma

converges to the continuous one as the meshsize tends to zero, and propose a discrete

gradient flow to compute discrete minimizers. Numerical experiments confirm the ability

of the scheme to approximate configurations with half-integer defects, and to deal with

colloidal and electric field effects. This work, joint with J.P. Borthagaray and S.

Walker, builds on our previous work for the Ericksen's model which we review briefly.

We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere. We prove that, in the low temperature regime, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into the sphere. This flow is a special case of the gradient flow to the Oseen-Frank energy functional for nematic liquid crystals and the existence of its global weak solution was first obtained by Y.M Chen, using Ginzburg-Landau approximation.  The key ingredient of our result is to show the strong compactness of the family of number density functions and the proof relies on the strong compactness of the corresponding second moment (or the Q-tensor), a spectral decomposition of the linearized operator near the limiting local equilibrium distribution, as well as the energy dissipation estimates.  This is a joint work with Wei Wang in Zhejiang university.
 

I will present a survey of commonly considered notions of negative curvature for groups, focused on generalising properties of Gromov hyperbolic groups.

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N=4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

In this talk I discuss how we can simulate stochastic chemical kinetics when there is a memory component. This can occur when there is spatial crowding within a cell or part of a cell, which acts to constrain the motion of the molecules which then in turn changes the dynamics of the chemistry. The counterpart of the Law of Mass Action in this setting is through replacing the first derivative in the ODE description of the Law of Mass Action by a time-­fractional derivative, where the time-­fractional index is between 0 and 1. There has been much discussion in the literature, some of it wrong, as to how we model and simulate stochastic chemical kinetics in the setting of a spatially-­constrained domain – this is sometimes called anomalous diffusion kinetics.

In this presentation, I discuss some of these issues and then present two (equivalent) ways of simulating fractional stochastic chemical kinetics. The key here is to either replace the exponential waiting time used in Gillespie’s SSA by Mittag-­Leffler waiting times (MacNamara et al. [2]), which have longer tails than in the exponential case. The other approach is to use some theory developed by Jahnke and Huisinga [1] who are able towrite down the underlying probability density function for any set of mono-­molecular chemical reactions (under the standard Law of Mass Action) as a convolution of either binomial probability density functions or binomial and Poisson probability density functions). We can then extend the Jahnke and Huisinga formulation through the concept of iterated Brownian Motion paths to produce exact simulations of the underlying fractional stochastic chemical process. We demonstrate the equivalence of these two approaches through simulations and also by computing the probability density function of the underlying fractional stochastic process, as described by the fractional chemical master equation whose solution is the Mittag-­Lefflermatrix function. This is computed based on a clever algorithm for computing matrix functions by Cauchy contours (Weideman and Trefethen [3]).

This is joint work with Manuel Barrio (University of Vallodolid, Spain), Kevin Burrage (QUT), Andre Leier (University of Alabama), Shev MacNamara(University of Technology Sydney)and T. Marquez-­Lago (University of Alabama).

[1]T. Jahnke and W. Huisinga, 2007, Solving the chemical master equation for monomolecular reaction systems analytically, J. Math. Biology 54, 1, 1—26.[2]S. MacNamara, B. Henry and W. McLean, 2017, Fractional Euler limits and their applications, SIAM J. Appl. Math. 77, 2, 447—469.[3]J.A.C. Weideman and L.N. Trefethen, 2007, Parabolic and hyperbolic contours for computing the Bromwich integral, Math. Comp. 76, 1341—1356.

ACORN generators represents an approach to generating uniformly distributed pseudo-random numbers which is straightforward to implement for arbitrarily large order $k$ and modulus $M=2^{30t}$ (integer $t$). They give long period sequences which can be proven theoretically to approximate to uniformity in up to $k$ dimensions, while empirical statistical testing demonstrates that (with a few very simple constraints on the choice of parameters and the initialisation) the resulting sequences can be expected to pass all the current standard tests .

The standard TestU01 Crush and BigCrush Statistical Test Suites are used to demonstrate for ACORN generators with order $8≤k≤25$ that the statistical performance improves as the modulus increases from $2^{60}$ to $2^{120}$. With $M=2^{120}$ and $k≥9$, it appears that ACORN generators pass all the current TestU01 tests over a wide range of initialisations; results are presented that demonstrate the remarkable consistency of these results, and explore the limits of this behaviour.

This contrasts with corresponding results obtained for the widely-used Mersenne Twister MT19937 generator, which consistently failed on two of the tests in both the Crush and BigCrush test suites.

There are other pseudo-random number generators available which will also pass all the TestU01 tests. However, for the ACORN generators it is possible to go further: we assert that an ACORN generator might also be expected to pass any more demanding tests for $p$-dimensional uniformity that may be required in the future, simply by choosing the order $k>p$, the modulus $M=2^{30t}$ for sufficiently large $t$, together with any odd value for the seed and an arbitrary set of initial values. We note that there will be $M/2$ possible odd values for the seed, with each such choice of seed giving rise to a different $k$-th order ACORN sequence satisfying all the required tests.

This talk builds on and extends results presented at the recent discussion meeting on “Numerical algorithms for high-performance computational science” at the Royal Society London, 8-9 April 2019, see download link at bottom of web page http://acorn.wikramaratna.org/references.html.

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work, we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.

The renormalized expectation value of the stress energy tensor (RSET) is an object of central importance in quantum field theory in curved space-time, but calculating this on black hole space-times is far from trivial.  The vacuum polarization (VP) of a quantum scalar field is computationally simpler and shares some features with the RSET.  In this talk we consider the properties of the VP for a massless, conformally coupled scalar field on asymptotically anti-de Sitter black holes with spherical, flat and hyperbolic horizons.  We focus on the effect of the different horizon curvature on the VP, and the role played by the boundary conditions far from the black hole.     

This is a report on work in progress with Jingyin Huang. The complement of an arrangement of linear hyperplanes in a complex vector space has a natural “Borel-Serre bordification” as a smooth manifold with corners. Its universal cover is analogous to the Borel-Serre bordification of an arithmetic lattice acting on a symmetric space as well as to the Harvey bordification of Teichmuller space. In the first case the boundary of this bordification is homotopy equivalent to a spherical building; in the second case it is homotopy equivalent to curve complex of the surface. In the case of a reflection arrangement the boundary of its universal cover is the “curve complex” of the corresponding spherical Artin group. By definition this is the simplicial complex of all conjugates of proper, irreducible, spherical parabolic subgroups in the Artin group. A cohomological method is used to show that the curve complex of a spherical Artin group has the homotopy type of a wedge of spheres.

The principal ideal theorem (1930) guaranteed that any number field K would embed into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will finish the proof of their cohomological result and thus fully justify how it settled the class field tower problem.

After discussing the need for implicit constitutive relations to describe the response of both solids and fluids, I will discuss applications wherein such implicit constitutive relations can be gainfully exploited. It will be shown that such implicit relations can explain phenomena that have hitherto defied adequate explanation such as fracture and the movement of cracks in solids, the response of biological matter, and provide a new way to look at numerous non-linear phenomena exhibited by fluids. They provide a totally new and innovative way to look at the problem of Turbulence. It also turns out that classical Cauchy and Green elasticity are a small subset of the more general theory of elastic bodies defined by implicit constitutive equations. 

We discuss some results on integration of ``rough differential forms'', which are generalizations of classical (smooth) differential forms to similar objects involving Hölder continuous functions that may be nowhere differentiable. Motivations arise mainly from geometric problems related to irregular surfaces, and the techniques are naturally related to those of Rough Paths theory. We show in particular that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions (of Stratonovich or Ito type) and by summing over a refining sequence of partitions, leading to a two-dimensional extension of the classical Young integral, that coincides with the integral introduced recently by R. Züst. We further show that Stratonovich sums gives an advantage allowing to weaken the requirements on Hölder exponents, and discuss some work in progress in the stochastic case. Based on joint works with E. Stepanov, G. Alberti and I. Ballieul.

For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).
 

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as high-temperature limits of thermal states in many-body quantum mechanics.

In our work, we apply a perturbative expansion in the interaction. This expansion is then analysed by means of Borel resummation techniques. In two and three dimensions, we need to apply a Wick-ordering renormalisation procedure. Moreover, in one dimension, our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is based partly on joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

Moduli spaces of polarised varieties (varieties together with an ample line bundle) are not Hausdorff in general. A basic goal of algebraic geometry is to construct a Hausdorff moduli space of some nice class of polarised varieties. I will discuss how one can achieve this goal using the theory of canonical Kähler metrics. In addition I will discuss some fundamental properties of this moduli space, for example the existence of a Weil-Petersson type Kähler metric. This is joint work with Philipp Naumann.

I will talk about supersymmetric index of 4d N=1 supersymmetric theories on S^1xM_3 which counts supersymmetric states.  
In the first part, I will discuss a general formula to describe an asymptotic behaviour of the index in the limit of shrinking S^1
which we refer to as 4d (refined) supersymmetric Cardy formula. This part is based on arXiv:1611.00380 with Lorenzo Di Pietro.
In the second part, I will apply this formula to black hole physics. I will mainly focus on superconformal index of SU(N) N=4 super Yang-Mills theory
which is expected to be dual to type IIB superstring theory on AdS_5 x S^5. We will see that the index in the large-N limit reproduces the Bekenstein-Hawking entropy
of rotating charged BPS black hole on the gravity side. Our result for finite N makes a prediction to the black hole entropy with full quantum corrections.
The second part is based on arXiv:1901.08091.

Solidification processes of liquid metal alloys,  bubble dynamics (as in Taylor flows), pinch-offs of liquid-liquid jets, the formation of polymeric membranes, or the structure of high concentration photovoltaic cells are described by the dynamics of multiphase fluids. On the other hand, in applications such as mass spectrometry, lab-on-a-chip, and electro-fluidic displays, fluids on the micro-scale associated with a dielectric medium are of interest. Moreover, in many of these applications one is interested in influencing (or controlling) the underlying phenomenon in order to reach a desired goal. Examples for the latter could be the porosity structure of a polymeric membrane to achieve certain desired filtration properties of the membrane, or to optimize a microfluidic device for the transport of pharmaceutical agents.

A promising mathematical model for the behavior of multiphase flows associated with the applications mentioned above is given by a phase-field model of Cahn-Hilliard / Navier-Stokes (CHNS) type. Some strengths of phase field (or diffuse interface) approaches are due to their ability to overcome both, analytical difficulties of topological changes, such as, e.g., droplet break-ups or the coalescence of interfaces, and numerical challenges in capturing the interface dynamics between the fluid phases. Deep quenches in solidification processes of liquid alloys or rapid wall hardening in the formation of polymer membranes ask for non-smooth energies in connection with Cahn-Hilliard models. Analytically, this gives rise to a variational inequality coupled to the equations of hydrodynamics, thus yielding a non-smooth system (in the sense that the map associated with the underlying operator equation is not necessarily Frechet differentiable). In contrast to phase-field approaches,
one may consider sharp interface models. In view of this, our microfluidic applications alluded to above are formulated in terms of  sharp interface models and Hele-Shaw flows. In this context, we are particularly interested in applications of electrowetting on dielectric (EWOD) with contact line pinning. The latter phenomenon resembles friction, yields a variational inequality of the second kind, and – once again – it results in an overall nonsmooth mathematical model of the physical process.

   In both settings described above, optimal control problems are relevant in order to influence the underlying physical process to approach a desired system state.  The associated optimization problems are delicate as the respective constraints involve non-smooth structures which render the problems degenerate and prevent a direct application of sophisticated tools for the characterization of solutions. Such characterizations are, however, of paramount importance in the design of numerical solution schemes.

This talk addresses some of the analytical challenges associated with optimal control problems involving non-smooth structures, offers pathways to solutions, and it reports on numerical results for both problem classes introduced above.